Sergei Yakovenko's blog: on Math and Teaching

Monday, February 6, 2017

Lecture 12 (Jan 23, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 4:51
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Lie groups and Lie algebras

A Lie group is a smooth manifold with carries on it the structure of a group which is compatible with the smooth structure (i.e., the multiplication by an element of the group is a smooth self-map, necessarily a diffeomorphism, of the manifold).

This group structure means very high “homogeneity” of the manifold, in particular, existence of a flat connexion. On the other hand, there is a distinguished point on the manifold, corresponding to the group unit.

It turns out that the tangent space at the group unit is equipped with a natural operation, the antisymmetric bilinear bracket, closely related to the commutator of vector fields on the Lie group. This algebraic structure is called the Lie algebra, and it in a sense “encodes” the group structure.

The notes will be available later.

Wednesday, February 1, 2012

Last lecture: January 31, 2012

Integral: antiderivative and area

The last lecture (only partially exposed in the class) deals with the two seemingly unrelated problem: how to antidifferentiate functions (i.e., how to find a function when its derivative is known) and how to compute areas, in particular, under the graph of a given nonlinear function.

The answers turn out to be closely related by the famous Newton-Leibniz formula, which expresses the undergraph area through the antiderivative (primitive) of the function.

We discuss some tricks which allow to read the table of the derivatives from right to left (how to invert the Leibniz rule?) and find out that not all anterivatives can be “explicitly computed”. This “non-computability”, however, has its bright side: among “non-computable” antiderivatives we find functions which possess very special and useful properties, like the primitive of the power x^{-1}=\frac1x, which transforms multiplication into addition.

The lecture notes are available here.

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